CDS 101: Lecture 7.1 Loop Analysis of Feedback Systems

Transcription

1 CDS : Lct 7. Loop Analsis of Fback Sstms Richa M. Ma Goals: Show how to compt clos loop stabilit fom opn loop poptis Dscib th Nqist stabilit cition fo stabilit of fback sstms Dfin gain an phas magin an tmin it fom Nqist an Bo plots Raing: Åstöm an Ma, Analsis an Dsign of Fback Sstms, Ch 7 Avanc: Lwis, Chapt 7 Lct Rviw 6.: fom Tansf Last Fnctions Wk = Asin( ωt) x& = Ax B = Cx D x() = ss = H( jω) A ( ωt H jω ) sin ( ) Magni t 5 5 H () s = C( si A) B D nn H = H H = f Σ x&& x& x m s s C(s) b k Nov Nov R. RMM M. Ma, an HM, Caltch CDS

2 C(s) Clos Loop Stabilit Q: how o opn loop namics affct th clos loop stabilit? Givn opn loop tansf fnction C(s) tmin whn sstm is stabl Bt foc answ: compt pols clos loop tansf fnction H PC nn p c = = PC n n p c p c Altnativ: look fo conitions on PC that la to instabilit Exampl: if PC(s) = fo som s = jω, thn sstm is not asmptoticall stabl Conition on PC is mch nic bcas w can sign PC(s) b choic of C(s) Howv, chcking PC(s) = is not nogh; n mo sophisticat chck Pols of H = zos of PC Eas to compt, bt not so goo fo sign C(s) 6. Nov CDS Magnit (B) Gam Plan: Fqnc Domain Dsign Goal: fig ot how to sign C(s) so that C(s) is stabl an w gt goo pfomanc Pols of H PC = zos of PC H = Wol also lik to shap H PC to spcif pfomanc at iffnct fqncis Magnit (B) Bo Diagam PC PC 4 Low fqnc ang: PC PC PC (goo tacking) Banwith: fqnc at which clos loop gain = ½ opn loop gain Ia: s C(s) to shap PC (n ctain constaints) N tools to analz stabilit an pfomanc fo clos loop givn PC Nov CDS 4

3 C(s) Dtmin stabilit fom (opn) loop tansf fnction, L(s) = C(s). Us pincipl of th agmnt fom complx vaiabl tho (s aing) Thm (Nqist). Consi th Nqist plot fo loop tansf fnction L(s). Lt P # RHP pols of L(s) N # clockwis nciclmnts of Z # RHP zos of L(s) Thn Z = N P Nqist Cition To: Y() Nqist D conto Tak limit as, R Tac fom to along imagina axis Tac fqnc spons fo L(s) along th Nqist D conto Cont nt # of clockwis nciclmnts of th point Nov CDS j j Imag R Nqist Diagams Fom: U() jω < ω=j ω=j jω > Ral Axis ω= Ral N= Simpl Intptation of Nqist C(s) Basic ia: avoi positiv fback If L(s) has 8 phas (o gat) an gain gat than, thn signals a amplifi aon loop Us whn phas is monotonic Gnal cas qis Nqist Can gnat Nqist plot fom Bo plot flction aon al axis Bo Diagams Fom: U() Nqist Diagams Fom: U() ; Magnit (B) To: Y() Imagina Axis To: Y() ω= ω= ω= bo(ss) nqist(ss) Nov CDS 6 Ral Axis

4 Exampl: Popotional Intgal* sp contoll C(s) Imagina Axis To: Y() Nqist Diagams Fom: U() Ral Axis * slightl moifi; mo on th sign of this compnsato in nxt wk s lct / m Ps () = s b/ m s a Ki Cs () = Kp s. Rmaks N =, P = Z = (stabl) N to zoom in to mak s th a no nt nciclmnts Not that w on t hav to compt clos loop spons Nov CDS 7 Mo complicat sstms What happns whn opn loop plant has RHP pols? PC has singlaitis insi D conto ths mst b takn into accont Nqist Diagams Polzo map Fom: U() Imag Axis Imagina Axis To: Y() nstabl pol Ral Axis s Ls () = s 5 s s Ral Axis s = L ( s.5)( s.7. j)( s.7. j) N =, P = Z = NP = (stabl) Nov CDS 8 4

5 Commnts an cations Wh is th Nqist plot sfl? Ol answ: as wa to compt stabilit (bfo compts an MATLAB) Ral answ: givs insight into stabilit an obstnss; v sfl fo asoning abot stabilit Nqist plots fo sstms with pols on th jω axis H() s = ω= ss ( ) chos conto to avoi pols on axis n to cafll compt Nqist plot at ths points valat H(εj) to tmin iction ω=j ω=j ω= Cations with sing MATLAB MATLAB osn t gnat potion of plot fo pols on imagina axis Ths mst b awn in b han (mak s to gt th ointation ight!) Nov CDS 9 Rlativ stabilit: gain an phas magins Nqist plot tlls s if clos loop is stabl, bt not how stabl Gain magin How mch w can moif th loop gain an still hav th sstm b stabl Dtmin b th location wh th loop tansf fnction cosss 8 phas Phas magin How mch w can a phas la an still hav th sstm b stabl Dtmin b th phas at which th loop tansf fnction has nit gain Bo plot intptation Look fo gain =, 8 phas cossings MATLAB: magin(ss) Nqist Diagam Nov CDS ; Magnit (B) 5 Bo Diagam Gm=7.5 B (at.464 a/sc), Pm=8.754 g. (at.685 a/sc) 5 5

6 5 5 5 Nqist Diagam Ral Axis Nqist Diagam Ral Axis Exampl: cis contol / m Ps () = s b/ m s a C(s) Ki Cs () = Kp s. G(s) Gs () = Effct of aitional snso namics s Nw spomt has pol at s = (v fast); poblms vlop in th fil What s th poblm? A: insfficint phas magin in oiginal sign (not obst) 5 Bo Diagam Nqist plots Magnit (B) 5 Imagina Axis.5 Nqist Diagam Imagina Axis Ral Axis Nov CDS Pviw: contol sign / m Ps () = s b/ m s a C(s) Ki Cs () = α Kp s. G(s) Gs () = Appoach: Incas phas magin s Incas phas magin b cing gain can accommoat nw snso namics Taoff: low gain at low fqncis lss banwith, lag sta stat o 5 Bo Diagam Nqist plots Magnit (B) 5 α = 4 Imagina Axis Nqist Diagam Imagina Axis Ral Axis Nov CDS 6

7 5 5 Smma: Loop Analsis of Fback Sstms Bo Diagam Gm=7.5 B (at.464 a/sc), Pm=8.754 g. (at.685 a/sc) 5 C(s) Nqist citia fo loop stabilit Gain, phas magin fo obstnss ; Magnit (B) 5 j R Thm (Nqist). P # RHP pols of L(s) N # CW nciclmnts Z # RHP zos Nqist Diagam Z = N P j Nov CDS What s Nxt Homwok poblms Nqist plots; gain, phas magins Cis contol sign CDS : PI contol tim la Wnsa Nqist analsis Contol of scon o sstms Tim las Fia Hio Mabchi, Closloop atomic magntomt (qantm contol) Fia, pm, 74 JRG Nxt wk: PID Contol Dsign of contolls sing PID Rlativ stabilit an pfomanc Lct 8.: Fqnc Domain Dsign Loop Shaping fo Stabilit an Pfomanc Sta stat o, banwith, tacking Cs () Ls () Ps () Cs () Ls () Ps () Main ias Pfomanc spcifications giv bons on loop tansf fnction Us contoll to shap spons Gain/phas lationships constain sign appoach Stana compnsatos: popotional, la, PI 8 Nov CDS Don t fogt to fill ot MUD CARDS Nov CDS 4 7